4 Fractional Stationarity
[554.1.1]This section investigates the condition of invariance
or stationarity for the induced ultralong time dynamics
Sϖt*. [554.1.2]Invariance of a measure ν on G
under the induced dynamics Sϖt* is defined
as usual (see (3)) by requiring that
Sϖt*νB,t0*=νB,t0* |
| (22) |
for t>0 and B⊂G. [554.1.3]For 0<ϖ<1 (22) may be
called the condition of fractional invariance or
fractional stationarity. [554.1.4]Using (2) the
invariance condition becomes
[page 555, §0] for t>0 where Aϖ is the infinitesimal generator of the
semigroup Sϖt*. [555.0.1]For ϖ=1 the relation (21)
implies A1νB,t=-dνB,t/dt=0, and thus in this
case invariant measures conserve volumes in phase
space as usual. [555.0.2]A very different situation arises
for ϖ<1.
[555.1.1]For 0<ϖ<1 the infinitesimal generators of the stable convolution
semigroup Sϖt* are obtained [26] by evaluating the
generalized function s+-ϖ-1 [29] on the time translation
group Ts
Aϖϱt=c+∫0∞s-ϖ-1Ts-T0dsϱt=c+∫0∞s+-ϖ-1Tsdsϱt |
| (24) |
where c+>0 is a constant. [555.1.2]Comparing (24) with the
Balakrishnan algorithm [30, 31, 32] for
fractional powers of the generator of a semigroup Tt
-Aαϱt | = | limt→0+I-Tttαϱ |
| (25) |
| = | 1Γ-α∫0∞s-α-1I-Tsϱtds |
|
shows that if A=-d/dt denotes the infinitesimal generator of
the original time evolution Tt then
Aϖ=-Aϖ is the infinitesimal generator of the
induced time evolution Sϖt*. [555.1.3]For 0<ϖ<1 the generators Aϖ for Sϖt* are
fractional time derivatives [15, 31, 29]. [555.1.4]The differential
form (23) of the fractional invariance condition for ν becomes
for t>0 which was first derived in [18, 19]. [555.1.5]Its solution is
for t>0 with C0 a constant. [555.1.6]This shows that νB for a fractional
stationary dynamical state is not constant. [555.1.7]Fractional stationarity
or fractional invariance of a measure ν implies that
phase space volumes νB shrink with time. [555.1.8]Thus fractional
dynamics is dissipative. [555.1.9]More generally (26) reads
AϖνB,t=δt with solution νB,t=C0t+ϖ-1
for t≥0 in the sense of distributions. [555.1.10]The stationary solution
with ϖ=1 has a jump discontinuity at t=0, and is not simply
constant.
[555.2.1]The transition from an original invariant measure μ on Γ
to a fractional invariant measure ν on a subset G of measure
μG=0 may be called stationarity breaking. [555.2.2]It occurs
spontaneously in the sense that it is generated by the dynamics
itself. [555.2.3]Stationarity breaking implies ergodicity breaking, and
thus the ultralong time limit is a possible scenario for ergodicity
breaking in ergodic theory.
[555.3.1]The present paper has shown that the use of fractional time
derivatives in physics is not only justified, but arises
generically for induced dynamics in the ultralong time
limit. [555.3.2]This mathematical result applies to many physical
situations. [555.3.3]In the simplest case the resulting fractional
differential equation (26) defines fractional
stationarity which provides the dynamical basis for the
anequilibrium concept [18]. [555.3.4]Recently
fractional random walks were discussed [8]
and solved [10] in the continuum limit.
[page 556, §0] ACKNOWLEDGEMENT : The author is grateful to Norges
Forskningsrad (Nr.: 424.94 / 004 B) for financial support.